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Theorem splcl 12390
Description: Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Assertion
Ref Expression
splcl  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S splice  <. F ,  T ,  R >. )  e. Word  A )

Proof of Theorem splcl
Dummy variables  s 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2979 . . . 4  |-  ( S  e. Word  A  ->  S  e.  _V )
2 otex 4554 . . . 4  |-  <. F ,  T ,  R >.  e. 
_V
3 id 22 . . . . . . . 8  |-  ( s  =  S  ->  s  =  S )
4 fveq2 5688 . . . . . . . . . 10  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 1st `  b )  =  ( 1st `  <. F ,  T ,  R >. ) )
54fveq2d 5692 . . . . . . . . 9  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 1st `  ( 1st `  b
) )  =  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) )
65opeq2d 4063 . . . . . . . 8  |-  ( b  =  <. F ,  T ,  R >.  ->  <. 0 ,  ( 1st `  ( 1st `  b ) )
>.  =  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)
73, 6oveqan12d 6109 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. )  =  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) )
8 simpr 458 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  b  =  <. F ,  T ,  R >. )
98fveq2d 5692 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 2nd `  b
)  =  ( 2nd `  <. F ,  T ,  R >. ) )
107, 9oveq12d 6108 . . . . . 6  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. ) concat  ( 2nd `  b ) )  =  ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) )
11 simpl 454 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  s  =  S )
128fveq2d 5692 . . . . . . . . 9  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 1st `  b
)  =  ( 1st `  <. F ,  T ,  R >. ) )
1312fveq2d 5692 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 2nd `  ( 1st `  b ) )  =  ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) )
1411fveq2d 5692 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( # `  s
)  =  ( # `  S ) )
1513, 14opeq12d 4064 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >.  =  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. )
1611, 15oveq12d 6108 . . . . . 6  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( s substr  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. )  =  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) )
1710, 16oveq12d 6108 . . . . 5  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b ) ) >.
) concat  ( 2nd `  b
) ) concat  ( s substr  <.
( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. ) )  =  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >. ) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. ) ) )
18 df-splice 12230 . . . . 5  |- splice  =  ( s  e.  _V , 
b  e.  _V  |->  ( ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. ) concat  ( 2nd `  b ) ) concat  ( s substr  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. ) ) )
19 ovex 6115 . . . . 5  |-  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) )  e. 
_V
2017, 18, 19ovmpt2a 6220 . . . 4  |-  ( ( S  e.  _V  /\  <. F ,  T ,  R >.  e.  _V )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) ) )
211, 2, 20sylancl 657 . . 3  |-  ( S  e. Word  A  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) ) )
2221adantr 462 . 2  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) ) )
23 swrdcl 12311 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)  e. Word  A )
2423adantr 462 . . . 4  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)  e. Word  A )
25 ot3rdg 6592 . . . . . 6  |-  ( R  e. Word  A  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
2625adantl 463 . . . . 5  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
27 simpr 458 . . . . 5  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  R  e. Word  A )
2826, 27eqeltrd 2515 . . . 4  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( 2nd `  <. F ,  T ,  R >. )  e. Word  A )
29 ccatcl 12270 . . . 4  |-  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)  e. Word  A  /\  ( 2nd `  <. F ,  T ,  R >. )  e. Word  A )  -> 
( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) )  e. Word  A
)
3024, 28, 29syl2anc 656 . . 3  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) )  e. Word  A
)
31 swrdcl 12311 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. )  e. Word  A
)
3231adantr 462 . . 3  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. )  e. Word  A )
33 ccatcl 12270 . . 3  |-  ( ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) )  e. Word  A  /\  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. )  e. Word  A )  ->  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) )  e. Word  A )
3430, 32, 33syl2anc 656 . 2  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >. ) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. ) )  e. Word  A
)
3522, 34eqeltrd 2515 1  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S splice  <. F ,  T ,  R >. )  e. Word  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970   <.cop 3880   <.cotp 3882   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   0cc0 9278   #chash 12099  Word cword 12217   concat cconcat 12219   substr csubstr 12221   splice csplice 12222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-ot 3883  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-concat 12227  df-substr 12229  df-splice 12230
This theorem is referenced by:  psgnunilem2  15994  efglem  16206  efgtf  16212  frgpuplem  16262
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